Online robust locally differentially private learning for nonparametric regression

The growing prevalence of streaming data and increasing concerns over data privacy pose significant challenges for traditional nonparametric regression methods, which are often ill-suited for real-time, privacy-aware learning. In this paper, we tackle these issues by first proposing a novel one-pass online functional stochastic gradient descent algorithm that leverages the Huber loss (H-FSGD), to improve robustness against outliers and heavy-tailed errors in dynamic environments. To further accommodate privacy constraints, we introduce a locally differentially private extension, Private H-FSGD (PH-FSGD), designed to real-time, privacy-preserving estimation. Theoretically, we conduct a comprehensive non-asymptotic convergence analysis of the proposed estimators, establishing finite-sample guarantees and identifying optimal step size schedules that achieve optimal convergence rates. In particular, we provide practical insights into the impact of key hyperparameters, such as step size and privacy budget, on convergence behavior. Extensive experiments validate our theoretical findings, demonstrating that our methods achieve strong robustness and privacy protection without sacrificing efficiency.